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We derive a strong maximum principle for upper semicontinuous viscosity subsolutions of fully nonlinear elliptic differential equations whose dependence on the spatial variables may be discontinuous. Our results improve previous related ones for linear [18] and nonlinear [22] equations because we weaken structural assumptions on the nonlinearities. Counterexamples show that our results are optimal. Moreover they are complemented by comparison and uniqueness results, in which a viscosity subsolution is compared with a piecewise classical supersolution. It is curious to note that existence of a piecewise classical solution to a fully nonlinear problem implies its uniqueness in the larger class of continuous viscosity solutions.The purpose of this note is to derive a strong interior and boundary maximum principle for upper semicontinuous subsolutions of general nonuniformly elliptic equations of the form FxY uY DuY D 2 u 0 in WY 1 where W & R n is a connected domain and F is a Caratheodory function, i.e. F is measurable in x and continuous in the remaining variables. By a strong interior maximum principle we mean that u is constant if it attains its positive maximum at some interior point. The strong boundary maximum principle is also known as Hopfs second Lemma, see e.g. Lemma 3.4 in [8].It is well known [8,15,17] that the classical maximum principle holds for classsical solutions to elliptic equations of type (1), i) when these equations are uniformly elliptic, i.e. when there exist constants l K depending on a bound K on rY pY X and Y such thatii) when the function F is Lipschitz continuous in each variable, iii) and when FxY zY pY X is (nonstrictly) monotone increasing in z.Mathematics Subject Classification (1991): 35B05, 35B50, 35J60, 35R05.

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In this paper we investigate the Dirichlet problem \Ω is bounded. We sharpen previous non-existence results for this exterior Dirichlet problem by showing that even the smallness of the α-Holder norm, 0 < α < \ is not enough for the classical solvability of (1) and (2), not imposing any asymptotical conditions at infinity upon possible solutions. In particular, we explicitely exhibit smooth data / of arbitrary small C α -norm for which (1), (2) is not solvable in the space C°(Ω)ΠC 2 (Ω). The key idea of our proof is to replace the original problem (1), (2) on a known domain but with unknown boundary conditions at infinity by the corresponding problem on some unknown (bounded) domain, but with fixed boundary data. By the same method we show the instability of the exterior Dirichlet problem with respect to C a -small perturbations of the boundary data, 0